A336476 -id:A336476 - cp's OEIS frontend

A336648 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336476(i) = A336476(j), for all i, j >= 1.

1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 3, 1, 2, 1, 2, 2, 4, 2, 2, 2, 1, 2, 4, 2, 2, 3, 2, 1, 3, 2, 4, 1, 2, 2, 4, 2, 2, 4, 2, 2, 5, 2, 2, 2, 6, 1, 3, 2, 2, 4, 4, 2, 4, 2, 2, 3, 2, 2, 2, 1, 4, 3, 2, 2, 3, 4, 2, 1, 2, 2, 2, 2, 4, 4, 2, 2, 1, 2, 2, 4, 4, 2, 3, 2, 2, 5, 7, 2, 4, 2, 8, 2, 2, 6, 5, 1, 2, 3, 2, 2, 9

Offset: 1

Antti Karttunen, Jul 31 2020

nonn

Restricted growth sequence transform of A336476.

For all i, j: A324400(i) = A324400(j) => A003602(i) = A003602(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000593, A336475, A336476.

Cf. also A003602, A324400, A336320.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A000593(n) = sigma(n>>valuation(n, 2));
A336475(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,(1+f[i,2]) * (f[i,1]^f[i,2]))); };
A336476(n) = gcd(A000593(n), A336475(n));
v336648 = rgs_transform(vector(up_to,n,A336476(n)));
A336648(n) = v336648[n];

A336846 a(n) = gcd(sigma(A003961(n)), A000005(n)*A003961(n)).

Original entry on oeis.org

1, 2, 2, 1, 2, 12, 2, 4, 1, 4, 2, 6, 2, 12, 4, 1, 2, 2, 2, 2, 4, 4, 2, 120, 3, 12, 4, 6, 2, 24, 2, 2, 4, 4, 4, 1, 2, 12, 4, 8, 2, 24, 2, 26, 2, 12, 2, 6, 1, 6, 20, 18, 2, 24, 28, 24, 4, 4, 2, 12, 2, 4, 6, 1, 4, 24, 2, 2, 20, 24, 2, 20, 2, 12, 6, 6, 4, 24, 2, 2, 1, 4, 2, 36, 4, 12, 4, 8, 2, 4, 4, 6, 4, 12, 4, 12, 2, 2, 2, 3

Offset: 1

Antti Karttunen, Aug 06 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000005, A000203, A000290 (positions of odd terms), A003961, A003973, A324121, A336476, A336841, A336845, A336847, A336848.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336846(n) = { my(u=A003961(n),s=sigma(u)); gcd(s, numdiv(n)*u); };

a(n) = gcd(A003973(n), A336845(n)) = gcd(A003973(n), A336841(n)).
a(n) = gcd(A000203(A003961(n)), A000005(n)*A003961(n)).
a(n) = A324121(A003961(n)).

A336475 Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = (e+1)*p^e.

Original entry on oeis.org

1, 1, 6, 1, 10, 6, 14, 1, 27, 10, 22, 6, 26, 14, 60, 1, 34, 27, 38, 10, 84, 22, 46, 6, 75, 26, 108, 14, 58, 60, 62, 1, 132, 34, 140, 27, 74, 38, 156, 10, 82, 84, 86, 22, 270, 46, 94, 6, 147, 75, 204, 26, 106, 108, 220, 14, 228, 58, 118, 60, 122, 62, 378, 1, 260, 132, 134, 34, 276, 140, 142, 27, 146, 74, 450, 38, 308, 156

Offset: 1

Antti Karttunen, Jul 30 2020

Dirichlet convolution of A000265 with itself, divided by A001511(n).

Although for all i, j: A003602(i) = A003602(j) => a(i) = a(j), it is not true that a(i) = a(j) => A003602(i) = A003602(j), because A038040 has a duplicate occurrence of a term on at least two odd positions: A038040(1875) = A038040(3125) = 18750.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000005, A000265, A001227, A001511, A001620, A003602, A038040, A336476.

f[p_, e_] := (e+1)*p^e; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2023 *)

A336475(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,(1+f[i,2]) * (f[i,1]^f[i,2]))); };

from sympy import divisor_count
def A336475(n): return (m:=n>>(~n&n-1).bit_length())*divisor_count(m) # Chai Wah Wu, Jul 13 2022

a(n) = A038040(A000265(n)).
a(n) = A000265(n) * A001227(n).
a(n) = (A000005(n) * A000265(n)) / A001511(n). [See the first comment]
From Amiram Eldar, Sep 21 2023: (Start)
Dirichlet g.f.: ((2^s - 2)^2/(4^s - 2^s)) * zeta(s-1)^2.
Sum_{k=1..n} a(k) ~ (n^2/12) * (2*log(n) + 4*gamma + 10*log(2)/3 - 1), where gamma is Euler's constant (A001620). (End)

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A336648 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336476(i) = A336476(j), for all i, j >= 1.

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A336846 a(n) = gcd(sigma(A003961(n)), A000005(n)*A003961(n)).

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A336475 Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = (e+1)*p^e.

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